Precalculus Examples

$f (x) = \sqrt{x + 1}$ , $[0, 3]$

Step 1

Write $f (x) = \sqrt{x + 1}$ as an equation.

$y = \sqrt{x + 1}$

Step 2

Substitute using the average rate of change formula.

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Step 2.1

The average rate of change of a function can be found by calculating the change in $y$ values of the two points divided by the change in $x$ values of the two points.

$\frac{f (3) - f (0)}{(3) - (0)}$

Step 2.2

Substitute the equation $y = \sqrt{x + 1}$ for $f (3)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt{(3) + 1}) - (\sqrt{(0) + 1})}{(3) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $3$ and $1$ .

$\frac{\sqrt{4} - \sqrt{0 + 1}}{3 - (0)}$

Step 3.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2}} - \sqrt{0 + 1}}{3 - (0)}$

Step 3.1.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{2 - \sqrt{0 + 1}}{3 - (0)}$

Step 3.1.4

Add $0$ and $1$ .

$\frac{2 - \sqrt{1}}{3 - (0)}$

Step 3.1.5

Any root of $1$ is $1$ .

$\frac{2 - 1 \cdot 1}{3 - (0)}$

Step 3.1.6

Multiply $- 1$ by $1$ .

$\frac{2 - 1}{3 - (0)}$

Step 3.1.7

Subtract $1$ from $2$ .

$\frac{1}{3 - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{1}{3 + 0}$

Step 3.2.2

Add $3$ and $0$ .

$\frac{1}{3}$